Abstract:
Threshold resonance arises on the lower bound of the continuous spectrum of a quantum waveguide (the Dirichlet problem for the Laplace operator), provided that for this value of the spectral parameter a nontrivial bounded solution exists which is either a trapped wave decaying at infinity or an almost standing wave stabilizing at infinity. In many problems in asymptotic analysis, it is important to be able to distinguish which of the waves initiates the threshold resonance; in this work we discuss several ways to clarify its properties. In addition, we demonstrate how the threshold resonance can be preserved by fine tuning the profile of the waveguide wall, and we obtain asymptotic expressions for the near-threshold eigenvalues appearing in the discrete or continuous spectrum when the threshold resonance is destroyed.
Bibliography: 60 titles.
This research was carried out under a state assignment of the Ministry of Science and Higher Education of the Russian Federation (project no. ФФНФ-2021-0006).
Citation:
S. A. Nazarov, “The preservation of threshold resonances and the splitting off of eigenvalues from the threshold of the continuous spectrum of quantum waveguides”, Sb. Math., 212:7 (2021), 965–1000
\Bibitem{Naz21}
\by S.~A.~Nazarov
\paper The preservation of threshold resonances and the splitting off of eigenvalues from the threshold of the continuous spectrum of quantum waveguides
\jour Sb. Math.
\yr 2021
\vol 212
\issue 7
\pages 965--1000
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Linking options:
https://www.mathnet.ru/eng/sm9426
https://doi.org/10.1070/SM9426
https://www.mathnet.ru/eng/sm/v212/i7/p84
This publication is cited in the following 3 articles:
D.I. Borisov, D.A. Zezyulin, “On Perturbation of Thresholds in Essential Spectrum under Coexistence of Virtual Level and Spectral Singularity”, Russ. J. Math. Phys., 31:1 (2024), 60
S. A. Nazarov, “Raspredelenie mod sobstvennykh kolebanii v plastine, zaglublennoi v absolyutno zhëstkoe poluprostranstvo”, Matematicheskie voprosy teorii rasprostraneniya voln. 53, Zap. nauchn. sem. POMI, 521, POMI, SPb., 2023, 154–199
D. I. Borisov, D. A. Zezyulin, “On the bifurcation of thresholds of the essential spectrum with a spectral singularity”, Diff Equat, 59 (2023), 278–282
Citation in format AMSBIB
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